Math 2.0

Potential theory Harmonic functions Theorem

Harnack's principle

Builds on Harmonic functionHarnack's inequalityDomain

Statement

Let D⊆ℝnD \subseteq \R^n be a domain, that is, a connected open set. Let (uk)(u_k) be a sequence of harmonic functions on DD with u1≤u2≤u3≤⋯u_1 \le u_2 \le u_3 \le \cdots at each point of DD. Then one of the following holds.

  1. uk→∞u_k \to \infty uniformly on each compact subset of DD.
  2. (uk)(u_k) converges uniformly on each compact subset of DD to a harmonic function uu on DD.

Remarks

Proof idea: the functions uk−u1u_k - u_1 are nonnegative and harmonic. Harnack's inequality compares the values of a nonnegative harmonic function at two points of a compact set by constants that do not depend on the function. So the increasing limit is either infinite everywhere or finite everywhere, and in the finite case the convergence is uniform on compact sets. A uniform limit of harmonic functions is harmonic, by the mean value property. Connectedness is needed: on two disjoint discs, take uk=0u_k = 0 on one disc and uk=ku_k = k on the other.

Going deeper

For a student

The principle is the harmonic analogue of the monotone convergence theorem, and it follows from Harnack's inequality. It is used in Perron's method for the Dirichlet problem, where the solution is built as a limit of subharmonic functions. It also appears in the construction of Green's functions and harmonic measures by exhaustion of a domain.

Reading path

To understand this page from the ground up, read these 47 pages in order. The foundations come first.

  1. Element definition
  2. Euclidean space object
  3. Function concept
  4. Proposition concept
  5. Unit disc object
  6. Set definition
  7. Fréchet derivative definition
  8. Total derivative concept
  9. Function of several variables concept
  10. Vector-valued function concept
  11. Implication definition
  12. Predicate definition
  13. Ordered pair definition
  14. Limit of a function of several variables concept
  15. Differentiable function of several variables concept
  16. Vector field concept
  17. Partial derivative concept
  18. Existential quantifier definition
  19. Universal quantifier definition
  20. Cartesian product definition
  21. Continuity in R^n definition
  22. Differentiable function of several variables definition
  23. Partial derivative definition
  24. Subset definition
  25. Gradient concept
  26. Divergence concept
  27. Uniqueness quantifier definition
  28. Binary relation definition
  29. Differentiable on an open set definition
  30. Gradient definition
  31. Higher partial derivatives definition
  32. Jacobian matrix definition
  33. Laplacian concept
  34. Iterated derivative definition
  35. Hessian matrix definition
  36. Function definition
  37. Harmonic function concept
  38. C^k function characterised by continuous derivatives statement
  39. Domain definition
  40. Poisson kernel definition
  41. Vector field definition
  42. Harnack's inequality statement
  43. Divergence definition
  44. Laplacian definition
  45. Laplacian as a sum of second partial derivatives statement
  46. Harmonic function definition
  47. Harnack's principle statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Harnack%27s_principle
About this page
TagP15K8RKHDV
URIm2://P15K8RKHDV
Kindstatement, theorem
WritingW0 stub
MSC31B05, 31A05
StatusThis copy of the site has no build data. Run m2 status P15K8RKHDV.
AboutHarmonic function
UsesHarmonic function, Harnack's inequality, Domain
Same as
Changed2 November 2026

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